#152476

order	512
nilpotency class	5
isoclinism	isoclinic class of #13677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2196
commutator	#124	C₂²×C₄
Frattini, cosocle	#532	C₂³
socle	C₂	#12964
prime radical	#3150	C₂²
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

357 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,8,13,7,26)(3,14,18,42,19,44,17,43)(4,10,21,34,22,36,27,35)(5,11,25,39,24,38,28,40)(9,29,31,51,32,50,56,16)(15,45,47,41,48,62,52,63)(20,33,54,37,53,61,23,55)(30,57,59,64,60,49,58,46);(1,3,6,18,8,19,7,17)(2,9,12,31,13,32,26,56)(4,15,21,47,22,48,27,52)(5,16,25,29,24,51,28,50)(10,30,34,59,36,60,35,58)(11,14,39,42,38,44,40,43)(20,46,54,57,53,64,23,49)(33,45,37,41,61,62,55,63);(1,4)(2,10)(3,15)(5,23)(6,21)(7,27)(8,22)(9,30)(11,37)(12,34)(13,36)(14,41)(16,49)(17,52)(18,47)(19,48)(20,25)(24,54)(26,35)(28,53)(29,46)(31,59)(32,60)(33,40)(38,55)(39,61)(42,62)(43,45)(44,63)(50,64)(51,57)(56,58)

Polycyclic presentation:

2^9|a^2=f,b^2=f,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ehi,g^a=gh,e^b=eghi,h^b=hi,d^c=dg,e^c=ehi,g^c=gi,e^d=eh,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59672);